Logic Programming

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Logic Programming

• James Cheney
• CS 411

2
Functional Programming

• Programs as functions
• Write down function you want to calculate
• Computer evaluates to a value
• Based on lambda calculus, higher-order logic
• Examples LISP/Scheme, ML

3
Declarative Programming

• Programs as relations
• Write down a logical description of problem
• Computer searches for answer
• Based on first-order logic
• Best-known example Prolog

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Example 1 Append

• In ML
• fun append l l
• append (xxs) l
• x(append xs l)
• append(1,2,3,4)
• val it int list 1,2,3,4

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Example 1 Append

• In Prolog
• append(,L,L).
• append(XL,M,XN) -
• append(L,M,N).
• ?- append(1,2,3,4, X).
• X ? 1,2,3,4

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Reversibility

• Unlike ML programs, Prolog relations can be
  evaluated in reverse
• ?- append(X,3,4,1,2,3,4).
• X ? 1,2
• ?- append(1,X,3,4,1,2,3,4).
• X ? 2

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Non-determinsm

• Unlike ML, Prolog programs can be
  non-deterministic
• User can ask for more solutions by typing
• ?- append(X,Y,1,2,3,4).
• X -gt , Y -gt 1,2,3,4
• X -gt 1, Y -gt 2,3,4
• ...
• ?- append(X,X,1,2,3,4).
• no

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Types

• Prolog is untyped
• ?- append(,5,X).
• X ? 5
• ?- append(5,,X).
• no

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Terminology

• Terms
• constants c,
• function applications f(t1,t2,...),
• variables X (uppercase)
• lists , t1,t2,...
• Everything is a term

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Terminology

• Facts atomic predicates on terms
• append(t1,t2,t3)
• Clauses facts or rules of the form
• R(t1,...) - R1(t11,...),
• ...,
• Rn(tn1,...).

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Terminology

• Goal/query instance of relations
• append(1,2,3,4,X)
• Solution substitution answering goal
• X -gt 1,2,3,4

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Execution Model

• A Prolog program is a set of clauses followed by
  a goal.
• The program is run by trying to find a solution
  to the goal using the clause
• There may be zero, one, or many solutions

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Depth-first search

• Idea To solve goal G,
• If G is a clause in the program, then done
• Else, if G - G1,...,Gn is a clause in the
  program, solve G1 ... Gn.
• Else, give up on this goal.

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Append example

• append(,L,L).
• append(XL,M,XN) - append(L,M,N).

append(1,2,3,4,X)?
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Append example

• append(,L,L).
• append(XL,M,XN) - append(L,M,N).

append(1,2,3,4,X)?
X1,L2,M3,4,AXN
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Append example

• append(,L,L).
• append(XL,M,XN) - append(L,M,N).

append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,AXN
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Append example

• append(,L,L).
• append(XL,M,XN) - append(L,M,N).

X2,L,M3,4,N2N
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,A1N
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Append example

• append(,L,L).
• append(XL,M,XN) - append(L,M,N).

append(,3,4,N)?
X2,L,M3,4,N2N
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,A1N
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Append example

• append(,L,L).
• append(XL,M,XN) - append(L,M,N).

append(,3,4,N)?
L 3,4, N L
X2,L,M3,4,N2N
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,A1N
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Append example

• append(,L,L).
• append(XL,M,XN) - append(L,M,N).

Answer A 1N 12N
12L 1234
append(,3,4,N)?
L 3,4, N L
X2,L,M3,4,N2N
append(2,3,4,N)?
append(1,2,3,4,X)?
X1,L2,M3,4,A1N
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Matching goals and clauses

• Usually, when solving G with clause
• G ? D, G and G not exactly the same
• But through clever instantiations of variables,
  can make G G
• Example f(1,X) f(X,1) if X 1

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Exercises

• Write a Prolog program that reverses a list
• Solve the following for X and Y
• f(z,g(X,X)) f(X,Y)
• g(X,Y,z) g(Y,z,X)
• f(f(X,X),Y) f(f(f(Y),f(z)),w)